Detecting a stochastic background of gravitational waves in the presence of non-Gaussian noise: A performance of generalized cross-correlation statistic

We discuss a robust data analysis method to detect a stochastic background of gravitational waves in the presence of non-Gaussian noise. In contrast to the standard cross-correlation (SCC) statistic frequently used in the stochastic background searches, we consider a generalized cross-correlation (GCC) statistic, which is nearly optimal even in the presence of non-Gaussian noise. The detection efficiency of the GCC statistic is investigated analytically, particularly focusing on the statistical relation between the false-alarm and the false-dismissal probabilities, and the minimum detectable amplitude of gravitational-wave signals. We derive simple analytic formulas for these statistical quantities. The robustness of the GCC statistic is clarified based on these formulas, and one finds that the detection efficiency of the GCC statistic roughly corresponds to the one of the SCC statistic neglecting the contribution of non-Gaussian tails. This remarkable property is checked by performing the Monte Carlo simulations and successful agreement between analytic and simulation results was found.

Figure 1

Left: Probability distribution function of instrumental noise given by Eq. (9). The model parameters are set to ${\sigma }_{\mathrm{m}}=1$, ${\sigma }_{\mathrm{t}}=4$ and $P=0.1$. Here we dropped the detector label $i$. Right: Time-series data of the non-Gaussian noise generated by Eq. (9). Top panel shows the result with tail fraction $P=0.01$, while the bottom panel plots the case with larger value, $P=0.1$. In both panels, we specifically set the model parameters ${\sigma }_{\mathrm{m}}$ and ${\sigma }_{\mathrm{t}}$ as ${\sigma }_{\mathrm{m}}=1$ and ${\sigma }_{\mathrm{t}}=4$. For comparison, we also plot the weak signal of the stochastic gravitational waves with $ϵ=0.1$.

Figure 2

Derivative of the function $f(x)$ in the two-component Gaussian noise model. Left panel shows the dependence of the tail fraction $P_i$ keeping the ratio of noise variance fixed, i.e., ${\sigma }_{\mathrm{t},i}/{\sigma }_{\mathrm{m},i}=4$. On the other hand, right panel presents the dependence of the ratio ${\sigma }_{\mathrm{t},i}/{\sigma }_{\mathrm{m},i}$ keeping the tail fraction fixed, i.e., $P_i=0.1$.

Figure 3

The quantity ${\rho }_{\mathrm{eff}}/\rho$ defined in Eq. (29) as function of ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}$ in the case of the two identical detectors. From top to bottom, the tail fraction $P$ is chosen as $P=0.2$, 0.1, 0.05 and 0.01.

Figure 4

Analytic $P_{\mathrm{FA}}-P_{\mathrm{FD}}$ curves for the standard and generalized cross-correlation statistics in the presence of the non-Gaussian noises described by the specific model (9). The sold (dashed) lines represent the $P_{\mathrm{FA}}-P_{\mathrm{FD}}$ curves for the GCC (SCC) statistic for the stochastic signals with amplitude $ϵ=0.03$, 0.06 and 0.12 (top to bottom). Here, we assume that the two detectors are identical (see Eq. (31)). For each curves, the parameters are set as $P=0.01$, ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}=4$ and $N={10}^4$.

Figure 5

The function $G$ plotted against the ratio ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}$ in the case of two identical detectors (see Eq. (31)). Here, the tail fraction $P$ is specifically chosen as $P=0.01$, 0.05, 0.1 and 0.2 from top to bottom. The thick and thin lines are the function $G$ defined in Eq. (34) and the one taking account of the higher-order terms (a9), respectively.

Figure 6

$P_{\mathrm{FA}}-P_{\mathrm{FD}}$ curves for the GCC (left) and the SCC (right) statistics. Symbols denote the simulation results, while the lines indicate the analytic prediction from Eq. (29). In each panel, the ratio of the noise variance is fixed to ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}=4$ and the amplitude of stochastic signal is set to $ϵ=0.1$. Note that for the tail fraction $P=0.0$, corresponding to the Gaussian noise case, the solid line and the filled circles in left panel are identical to the one in right panel: $P=0.0$ (filled circles and solid); $P=0.05$ (open circles and dotted); $P=0.2$ (filled squares and dashed). The thin dashed line for $P=0.2$ indicates the analytic $P_{\mathrm{FA}}-P_{\mathrm{FD}}$ curve taking account of the higher-order terms (a9).

Figure 7

Same as in Fig. 6, but we here plot the dependence on the ratio ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}$, fixing the tail fraction and the amplitude of stochastic signals to $P=0.1$ and $ϵ=0.1$: ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}=2$ (filled circles and solid); ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}=4$ (open circles and dotted); ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}=8$ (filled squares and short-dashed); ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}=16$ (open squares and long-dashed).

Figure 8

Minimum detectable amplitude of the gravitational-wave signals as function of the tail fraction $P$ (left) and the ratio of noise variance ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}$ (right). The ratio of noise variance in left panel is specifically chosen as ${\sigma }_{\mathrm{t}}/{\sigma }_{\mathrm{m}}=4$, while the tail fraction in right panel is set to $P=0.1$. In both panels, filled (open) circles represent the simulation results derived from the GCC (SCC) statistic. The corresponding analytic curves are also shown in solid and dotted lines based on the expressions (32, 33). The thin line in left panel is the analytical prediction including the higher-order terms (a9). Note that in these plots, detection point is specifically set to $(P_{\mathrm{FA}}^{*},P_{\mathrm{FD}}^{*})=(0.1,0.1)$ with sample points $N={10}^4$.